**Dynamically Linked **

**Two Dimensional/ **

**One-Dimensional **

**Hydrodynamic Modelling **

**Program for Rivers, Estuaries **

**and Coastal Waters **

Prepared For:

Prepared By: WBM Oceanics Australia

**Offices ***Brisbane *
*Denver *
*Karratha *
*Melbourne *
*Morwell *
*Newcastle *
*Sydney *
*Vancouver*

**DOCUMENT CONTROL SHEET **

* Document: * WJS Thesis (Syme 1991).doc

* Title:* Dynamically Linked Two-Dimensional/

One-Dimensional Hydrodynamic Modelling
Program for Rivers, Estuaries & Coastal
Waters
**Project Manager: ****Author: ****Client:****Client Contact: ****Client Reference: **

**WBM Oceanics Australia**

Brisbane Office:
WBM Pty Ltd
Level 11, 490 Upper Edward Street
SPRING HILL QLD 4004
Australia
PO Box 203
Spring Hill QLD 4004
Telephone (07) 3831 6744
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**Synopsis: ****REVISION/CHECKING HISTORY**
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**DATE** **CHECKED BY** **ISSUED BY **

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**C**

**ONTENTS**

**Contents ** **i**

**List of Figures ** **iv**

**Summary ** **vi**

**List of Symbols ** **viii**

**Acknowledgements ** **xii**

**Qualification ** **xiii**

**1**

**I**

**NTRODUCTION**

**1-1**

**1.1** **General ** **1-1**

**1.2** **Benefits and Limitations of Dynamically Linked 2-D/1-D Models ** **1-4**

**1.3** **Available Computer Software ** **1-5**

**1.4** **Study Objectives ** **1-6**

**1.5** **Thesis Layout ** **1-6**

**2**

**S**

**ELECTION OF**

**S**

**CHEMES**

**2-1**

**2.1** **Summary ** **2-1**

**2.2** **Base Considerations ** **2-1**

2.2.1 Finite Differences versus Finite Elements 2-1

2.2.2 Courant Number/Computation Timestep 2-1

2.2.3 Explicit versus Implicit Schemes 2-2

**2.3** **Linking 1-D and 2-D Schemes ** **2-3**

2.3.1 General 2-3

2.3.2 Literature Review 2-3

2.3.3 Comments on the Literature 2-5

2.3.4 Methodology of 2-D/1-D Dynamic Link 2-6

2.3.5 Computation Timestep 2-6

2.3.6 Courant Number Constraints 2-6

2.3.7 Length and Orientation of 2-D/1-D Interface 2-7

**2.4** **1-D Scheme of ESTRY ** **2-8**

2.4.1 General 2-8

*CONTENTS* **II**

**2.5** **Two-Dimensional Scheme Selection ** **2-10**

2.5.1 Summary 2-10

2.5.2 ADI Schemes 2-11

2.5.3 Other Schemes 2-13

**3**

**T**

**WO**

**-D**

**IMENSIONAL**

**H**

**YDRODYNAMIC**

**M**

**ODELLING**

**P**

**ROGRAM**

**3-1**

**3.1** **Summary ** **3-1**

**3.2** **Solution Scheme for the 2-D SWE ** **3-3**

3.2.1 Overview 3-3

3.2.2 2-D Shallow Water Equations (SWE) 3-3

3.2.3 Computation Procedure 3-4

3.2.4 Open Boundaries 3-5

3.2.5 Closed Boundaries 3-6

**3.3** **Coding Finite Difference Equations ** **3-7**

3.3.1 Overview 3-7

3.3.2 Stage One - Step One 3-9

3.3.3 Stage One - Step Two 3-12

**3.4** **Validation and Testing ** **3-16**

3.4.1 Summary 3-16

3.4.2 Uniform Flow 3-18

3.4.3 Diverging Flow 3-29

3.4.4 Flow Past a Solid Wall 3-32

3.4.5 S-Shaped Channel 3-46

**3.5** **Wetting and Drying ** **3-47**

3.5.1 General 3-47

3.5.2 No Free Overfall Method 3-48

3.5.3 Free Overfall Method 3-49

3.5.4 Test Results 3-53

**4**

**T**

**WO**

**D**

**IMENSIONAL**

**/O**

**NE**

**D**

**IMENSIONAL**

**D**

**YNAMIC**

**L**

**INK**

**4-1**

**4.1** **Summary ** **4-1**

**4.2** **Description of Method ** **4-2**

4.2.1 Boundary Definition 4-2

4.2.2 Methodology 4-3

4.2.3 Water Levels of 1-D & 2-D Models 4-4

**4.3** **Validation and Testing ** **4-5**

4.3.2 Uniform Flow 4-5

4.3.3 Dynamic Flow 4-9

**4.4** **Treatment of Oblique Boundaries ** **4-14**

4.4.1 General 4-14

4.4.2 Discharge Across an Oblique Boundary 4-14

4.4.3 Stabilising Oblique Water Level Boundaries 4-16

**5**

**C**

**ONCLUSION**

**5-1**

**6**

**R**

**EFERENCES**

**6-1**

*LIST OF FIGURES* **IV**

**L**

**IST OF **

**F**

**IGURES**

**Figure 1.1** **Linked 2-D/1-D Model Layout ** **1-3**

**Figure 2.1** **Spatial Grid of Coupled Model, (Figure 4.1, Stelling 1981) ** **2-4**
**Figure 2.2** ** A 1-D Element Linked to Multiple 2-D Grid Elements ** **2-8**

**Figure 2.3** ** 2-D/1-D Link at an Oblique Boundary ** **2-8**

**Figure 3.1** **Spatial Location of Variables Relative to a Grid Element ** **3-8**

**Figure 3.2** **Uniform Flow Model Layout ** **3-19**

**Figure 3.3a** **Water Level Time Histories - Uniform Flow - ?t = 15 s ** **3-21**
**Figure 3.3b** **Water Level Time Histories - Uniform Flow - ?t = 360 s ** **3-22**
**Figure 3.3c** **Flow Time Histories - Uniform Flow - ?t = 15 s ** **3-23**
**Figure 3.3d** **Flow Time Histories - Uniform Flow - ?t = 360 s ** **3-24**
**Figure 3.3e** **Flow (Fine Scale) Time Histories - Uniform Flow - ?t = 15 s ** **3-25**
**Figure3.3f** **Flow (Fine Scale) Time Histories - Uniform Flow - ?t = 360 s ** **3-26**
**Figure 3.3g** **Water Surface Profiles - Uniform Flow - ?t = 15 s ** **3-27**
**Figure 3.3h** **Water Surface Profiles - Uniform Flow - ?t = 360 s ** **3-28**

**Figure 3.4** **Diverging Flow Model Layout ** **3-29**

**Figure 3.5** **Time Histories and Water Level Profiles - Diverging Flow ** **3-31**

**Figure 3.6** **Flow Past a Solid Wall Model Layout ** **3-32**

**Figure 3.7** **Flow Patterns and Water Level Contours for ?t = 10 s Flow Past a Solid Wall 3-33**
**Figure 3.8a** **Water Level Time Histories for ?t = 1 s, 10 s and 30 s Flow Past a Solid Wall 3-34**
**Figure 3.8b** **Flow Patterns at 2 h for ?t = 1 s, 10 s and 30 s Flow Past a Solid Wall 3-35**
**Figure 3.9a** **Water Level Time Histories for µ = zero, 0.1, 1 and 10 m2/s Flow Past a Solid Wall**

**3-36**
**Figure 3.9b** **Flow Patterns and Change in Velocity Contours at 2 h for µ = zero, 0.1, 1 and 10 **

**m2/s - Flow Past a Solid Wall ** **3-37**

**Figure 3.10** **Arr 4 Model Layout - Flow Past a Solid Wall ** **3-38**
**Figure 3.11a** **Water Level Time Histories for Other Boundary Arrangements - Flow Past a **

**Solid Wall ** **3-40**

**Figure 3.11b** **Flow Time Histories for Other Boundary Arrangements Flow Past a Solid Wall**
**3-41**
**Figure 3.11c** **Flow Patterns at 2 h for Other Boundary Arrangements Flow Past a Solid Wall**

**3-42**
**Figure 3.11d** **Time Histories for 2-D/1-D Arrangement Flow Past a Solid Wall ** **3-43**
**Figure 3.11e** **Flow Patterns at 1, 2, 3 and 4 h for 2-D/1-D Arrangement Flow Past a Solid Wall**

**3-44**

**Figure 3.12** **S-Shaped Channel Model Layout ** **3-46**

**Figure 3.13** **Flow Patterns at Mid Flood Tide for Cr = 16 and 96 S-Shaped Channel 3-46**
**Figure 3.14** **Free Overfall Variables for a ****v**** Velocity Point ** **3-50**
**Figure 3.15** **Wetting and Drying Test Case 1 Model Layout ** **3-54**
**Figure 3.16a** **Water Level Profiles - Wetting and Drying Test Case 1 ** **3-55**

**Figure 3.16b** **Water Level Profiles - Comparison of Methods Wetting and Drying Test Case 1**
**3-56**
**Figure 3.16c** **Flow Patterns at 10 h - Comparison of Methods Wetting and Drying Test Case 1**

**3-57**
**Figure 3.16d** **Integral Flow Time Histories - Comparison of Methods Wetting and Drying Test **

**Case 1 ** **3-58**

**Figure 3.16e** **Integral Flow (Fine Scale) Time Histories Comparison of Methods - Wetting and **

**Drying Test Case 1 ** **3-59**

**Figure 3.17** **Wetting and Drying Test Case 2 Model Layout ** **3-60**
**Figure 3.18a** **Flow Patterns at 1 h and 3 h - Comparison of Methods Wetting and Drying Test **

**Case 2 ** **3-62**

**Figure 3.18b** **Flow Patterns at 5 h and 7 h - Comparison of Methods Wetting and Drying Test **

**Case 2 ** **3-63**

**Figure 3.18c** **Flow Patterns at 8 h and 8½ h - Comparison of Methods Wetting and Drying **

**Test Case 2 ** **3-64**

**Figure 3.18d** **Flow Patterns at 9 h and 10 h - Comparison of Methods Wetting and Drying **

**Test Case 2 ** **3-65**

**Figure 3.19** **Water Level Contours at 3½ and 10 h Comparison of Methods - Wetting and **

**Drying Test Case 2 ** **3-66**

**Figure 4.0** **Computation Cycle for 2-D/1-D Link over One Timestep ** **4-4**
**Figure 4.1** **Uniform Flow Test Model Layouts for 2-D/1-D Link ** **4-6**
**Figure 4.2a** **Flow Time Histories - 2-D/1-D Uniform Flow ** **4-7**
**Figure 4.2b** **Velocity Time Histories - 2-D/1-D Uniform Flow ** **4-8**
**Figure 4.2c** **Water Surface Profiles - 2-D/1-D Uniform Flow ** **4-9**
**Figure 4.3** **Dynamic Flow Model Layouts for 2-D/1-D Link ** **4-10**
**Figure 4.4a** **Flow Integral Time Histories - 2-D/1-D Dynamic Flow ** **4-11**
**Figure 4.4b** **Water Surface Profiles - 2-D/1-D Dynamic Flow ** **4-12**
**Figure 4.4c** **Change in Flow Patterns at Peak Ebb (top) & Flood Tide 2-D/1-D Dynamic Flow**

**4-13**
**Figure 4.5** **Modelled and Actual Water Level Boundary Lines ** **4-14**

**Figure 4.6** **Flow Across an Oblique Line ** **4-15**

**Figure 4.7** **Location of Velocity Components for Different Angles of Flow ** **4-15**
**Figure 4.8** **Weighting Functions Applied at Oblique Water Level Boundaries ** **4-18**
**Figure 4.9** **Oblique Boundary Test Model Layout - 2-D Section ** **4-20**
**Figure 4.10a** **Instability Generation at an Oblique Boundary ** **4-21**
**Figure 4.10b** **Flow Patterns at 6 h for Methods 1 and 2 Oblique Boundary Test ** **4-22**

*SUMMARY* **VI**

**S**

**UMMARY**

Hydrodynamic modelling practices are greatly enhanced by combining one-dimensional (1-D) and two-dimensional (2-D) models into one overall model. The benefits are: greater efficiency; reduced computation time; greater accuracy; impacts from works are registered in all models; and, improved user satisfaction and performance.

To realise these benefits, a dynamically linked 2-D/1-D hydrodynamic modelling program, codenamed TUFLOW, was developed. It is specifically designed for the modelling of long waves in rivers, estuaries and coastal waters.

For the 1-D component, the ESTRY program was used without any significant modification. ESTRY is an established program, based on an explicit finite difference method (FDM) for the 1-D shallow water equations (SWE). For the 2-D component, new computer code was written to emulate the 2-D SWE solution scheme proposed in Stelling (1984). The Stelling scheme is an alternating direction implicit (ADI) FDM based on the well-known Leendertse (1967, 1970) schemes.

The code written for the 2-D scheme was validated and assessed using a range of test models. The results from the testing found that:

1 Mass was conserved 100 % between internal grid elements.

2 The advection terms in the momentum equation were performing correctly.

3 Water level boundaries were extremely stable, but incorrect results can occur where only water level boundaries are specified, as the discharge across the model's boundaries is not defined. 4 Flow boundaries were highly unstable for steady state simulations, and where stable, caused a

loss of mass of up to 3 % for large Courant Numbers (> 50). They showed good stability in dynamic simulations.

Incorporated into TUFLOW are three significant developments, namely:

1 A method for wetting and drying of 2-D intertidal flats, which allows the continued draining of perched waters.

2 The dynamic linking of any number of ESTRY 1-D network models to a TUFLOW 2-D model. 3 The stabilisation of water level boundaries lying obliquely to a 2-D orthogonal grid, therefore,

providing maximum flexibility for locating the 2-D/1-D interface.

A comprehensive computer graphics system was also developed for the purpose of this study, but is not presented as part of this thesis. Aspects of the above developments have been briefly described in Syme (1989, 1990a), and Syme and Apelt (1990b).

The wetting and drying method developed is shown to calculate the tidal prism more accurately, and to eliminate the occurrence of surges when perched waters are re-connected with the rising tide. Application in practice has shown the method to be a significant improvement over convential methods, especially in models with large areas of intertidal flats.

For dynamically linking the 1-D and 2-D schemes, a simple but effective method is proposed, which utilises the excellent stability properties of the 2-D scheme's water level boundaries.

The 1-D and 2-D boundaries at the interface are treated as normal open boundaries. A flow boundary is specified for the 1-D model and a water level boundary for the 2-D model. At the end of each computation step boundary condition data is transferred from one model to the other. For the 1-D boundary, the flow across the 2-D model boundary is specified, while for the 2-D boundary the water level from the 1-D model is used.

The conditions at the 2-D/1-D interface are that the water level is approximately horizontal, there is negligible variation in the convective acceleration terms and the flow is perpendicular to the 2-D model boundary. These are essentially the same conditions which are applied at the open boundaries of the 1-D and 2-D schemes.

Testing has shown the 2-D/1-D link to conserve mass 100 % and to have no stability problems. Application to more than twenty production models has shown it to be a very useful and powerful facility.

A key aspect in the successful application of the 2-D/1-D link was the development of a method for stabilising water level boundaries which lie obliquely to the orthogonal 2-D grid. This provides maximum flexibility for locating the interface, which in practice is extremely important, because of the irregularity of natural water bodies.

This thesis presents a discussion on the process of selecting the 1-D and 2-D schemes; and the details of: the methodology used for coding the D scheme; the wetting and drying method; the dynamic 2-D/1-D link; and the stabilisation of oblique water level boundaries. Selected results from test cases are also presented.

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*Timestep*

*n*

*Computatio*

*=*

*t*

*bv*

*4*

*3*

*2*

*1*

*0*

*φ*

*µ*

*ζ*

*ζ*

*ζ*

*ζ*

*ζ*

*ζ*

*ζ*

*ζ*

*δ*∆ ∆ ∆

*ACKNOWLEDGEMENTS* **XII**

**A**

**CKNOWLEDGEMENTS**

The author wishes to acknowledge the following persons and organisation, who gave the support, encouragement and patience necessary for the completion of this study.

• WBM Pty Ltd (Oceanics Australia).

• Professor Colin Apelt (The University of Queensland).

• Dean Patterson, Craig Witt and Tony McAlister (whom the author can fully recommend for providing constructive suggestions and finding "bugs" in "user-friendly" software).

• Georgina Warrington.

**Q**

**UALIFICATION**

All that is presented and discussed in the content of this thesis, including any reference to computer code (except for the ESTRY program), was the work of the author.

*INTRODUCTION*

**1-1**

**1-1**

**1 **

**I**

**NTRODUCTION**

**1.1 **

**General **

Quantifying the hydraulic behaviour of natural and artificial watercourses is often essential for (a) understanding hydraulic, biological and other processes, (b) predicting impacts of works and natural events, and (c) environmental management. Numerical hydrodynamic models are presently the most efficient, versatile and widely used tools for these purposes. There are, however, opportunities for development, amelioration and refinement of the mathematical methodologies used.

A hydrodynamic model aims to represent numerically the topography and roughness of a watercourse, so as to be a sufficiently accurate approximation of the real situation. Along the boundaries of the model, estimates of the hydraulic conditions need to be specified prior to a simulation. These boundary conditions could be the water levels of an ocean tide or the flow in a river. A computer program simulates the hydraulic behaviour over time throughout the domain of the model using the free surface fluid flow equations.

The general purpose computer software used for developing, calculating and displaying the results of hydrodynamic models can be classified into two areas. Firstly, the coding for the solution scheme and, secondly, for processing the input and output data. There is a need for continued improvement and optimisation of the solution schemes so more complex and detailed hydrodynamic systems can be modelled. In conjunction with this, the development of user-friendly, menu driven, graphic environments has many benefits.

The development of computer graphic systems for data input and output is bein g driven by further advances in computer hardware. This must not be brushed aside as fancy trimming, but as the opportunity to develop easily and efficiently hydrodynamic models of great complexity, and to enhance the visualisation and understanding of model results. To realise these opportunities the development of computer software in this area is necessary and should be given a high priority. The solution schemes must be accurate, robust and versatile to be of use in practice. No one universal scheme has yet been developed because of a combination of factors such as the complexity and range of scale of natural waterways, limitations of computer hardware, and cost versus benefit considerations. Schemes are therefore varied in their complexity, accuracy and robustness, and are designed for specific areas of application.

The schemes are approximate solutions of the partial differential equations of mass and momentum conservation for unsteady free-surface flow of long waves. These equations are known as the shallow water equations (SWE), and are typically used for modelling floods, ocean tides or storm surges.

SWE schemes are categorised as either one-dimensional (1-D), two-dimensional (2-D) or three-dimensional (3-D). A 1-D scheme is based on a form of the 1-D SWE, a 2-D scheme on the 2-D SWE, and so on. 1-D schemes are the simplest and most economical, while 3-D schemes are complex and highly demanding on computational effort and computer memory. 1-D and 2-D schemes are by far the most widely used and are those which we are concerned with in this study.

1-D schemes are typically used for rivers and estuaries as the flow is essentially channelised or one-directional. In these schemes the water velocities are calculated as a cross-sectional average in the direction of flow. 1-D solution schemes are often networked to give a quasi 2-D representation and to model branching of water courses. These models are widely used for modelling the flooding of rivers and floodplains, and tidal flows in estuaries.

2-D schemes are normally in plan view with the velocities averaged over the depth of the water column. They are mostly used for coastal waters where the water moves in a horizontal plane and also in sections of rivers and estuaries where a more detailed study of the flow patterns is required. 3-D schemes are occasionally adopted where the study requires consideration of density and temperature variations in the water column or the flows are sufficiently complex to warrant the third dimension. 2-D schemes can be layered to give a quasi 3-D effect.

Clearly, the domains of 1-D and 2-D models must meet and overlap. For example, the river (1-D model) discharging into a coastal bay (2-D model), or when modelling on a fine scale the 2-D flow patterns in a section of an estuary.

Representing such scenarios as one overall model composed of a combination of 1-D and 2-D models, has numerous benefits relating to greater flexibility, accuracy, efficiency and performance. To realise these benefits a dynamic link between the 1-D and 2-D schemes is required. The dynamic link transfers hydrodynamic variables from one model to the other as the computation proceeds, thus removing the need to simulate 1-D and 2-D models separately. An example of how a dynamically linked 2-D/1-D model would be schematised is illustrated in Figure 1.1.

The 2-D/1-D dynamic link, despite its many benefits, is only mentioned in the literature a few times (Stelling 1981, Danish Hydraulic Institute 1983 and Crean et al 1988). The development of a robust and widely applicable 2-D/1-D dynamically linked scheme for use in practice does not appear to have been publicised. To realise the benefits of a 2-D/1-D link a study was initiated by the author, the results of which are presented herein.

*INTRODUCTION*

**1-3**

**1-3**

**Figure **
**1.1 Linked 2-D/1-D Model Layout **

**1.2 **

**Benefits and Limitations of Dynamically Linked **

**2-D/1-D Models **

The benefits of representing a hydrodynamic system as a combination of 1-D and 2-D models dynamically linked together are:

(a) greater flexibility and versatility when developing a hydrodynamic model,

(b) the hydrodynamic response resulting from a change to one model is registered in all other models (this is particularly important for assessing hydraulic impacts),

(c) greater computational efficiency and accuracy, and (d) user satisfaction and performance.

A more general discussion of the benefits is presented in point form below.

(i). Most hydrodynamic systems of coastal regions are best represented by a combination of 1-D and 2-D models. For example, in the case of a river (1-D model), flowing into a coastal bay (2-D model). In such situations, the traditional approach has been to use separate models. This can lead to uncertainties and inaccuracies if the boundary conditions of the models where they interface are not clearly defined. If the 1-D and 2-D models can be dynamically linked these uncertainties and inaccuracies would be eliminated.

(ii). A 2-D model of an estuary can often not cover the whole tidal waterway because of limitations on the model dimensions and computation time. This necessitates that the neglected waterways must be (a) represented by artificially created areas within the 2-D model domain, (b) represented by a separate 1-D model as discussed above, or (c) ignored. If significant areas of a tidal waterway are ignored or poorly represented, a tide model will produce inaccurate results as it is essential that the full tidal compartment be modelled. By being able to link 1-D models dynamically onto a 2-D model, the entire tidal waterway area outside the 2-D model domain can be simply and easily represented.

(iii). The majority of hydrodynamic modelling exercises are carried out to establish the impacts of works and natural events, and to provide engineering solutions to minim ise these impacts. If impacts extend outside the domain of a model, their full extent can not be assessed accurately. The ability to represent easily a greater area with dynamically linked 1-D and 2-D models would largely overcome this problem as the impacts would be registered in all models. This is not the case where separate 1-D and 2-D models are used.

(iv). The domain of a 2-D model may also have to be extended well away from the area of interest to define an open boundary adequately. If part of this extension of the 2-D model can be represented by a 1-D model, the coverage of the 2-D model can be reduced allowing decreased computational times and/or a finer spatial discretisation.

(v). The greater flexibility, accuracy and versatility associated with a 2-D/1-D dynamic link would provide indirect benefits of user satisfaction and performance. These benefits would result from

*INTRODUCTION*

**1-5**

**1-5**

those benefits already mentioned such as reduced data input; higher turnover of simulations because of faster computations; greater flexibility and greater confidence in the results.

Limitations of the 2-D/1-D dynamic link would be:

(a) Stability constraints at the 2-D/1-D interface which override those of either the 1-D or 2-D schemes.

(b) Loss of accuracy caused by assumptions built into the 2-D/1-D link. (c) Numerical disturbances caused by the 2-D/1-D link.

**1.3 **

**Available Computer Software **

Suitable computer software code which was freely available is briefly discussed in this section. The computer programs were developed by WBM Pty Ltd or The University of Queensland.

It is noted that software which resulted from this project was to be used for both consulting and research purposes. Some preliminary investigations showed that this essentially excluded software from other organisations as there was generally a major cost involved if it was to be used for consulting. Also some organisations only offer executable code, not source code as was required. The other consideration is the benefits gained from developing in-house software. This should not be over-looked, and was the preferred approach at all times.

**1-D Hydrodynamic Programs **

WBM Pty Ltd has developed and used extensively the 1-D hydrodynamic network program, ESTRY (Morrison and Smith, 1978) since 1974. It can be regarded as a proven program, having been widely used in practice, and scrutinised.

The University of Queensland has also developed a 1-D hydrodynamic program (Apelt, 1982). Although not widely used in design applications it has had many research applications and can be regarded as capable and accurate.

For the purposes of this project, ESTRY was given preference for the following reasons:

• the author's familiarity with the program coding and his involvement in ESTRY's development.

• ESTRY has had far wider use in practice with a large bank of developed and calibrated models available for use if required.

**2-D Hydrodynamic Programs **

WBM Pty Ltd has available the 2-D hydrodynamic modelling program, TWODIM, which is based on the well-known method described in Leendertse (1967). This method has found widespread use in the consulting and research fields because of the public availability of the computer code and Leendertse's good documentation. It has, however, a number of limitations relating to numerical stability and accuracy and is today regarded as somewhat outdated. During the 1980's a number of alternative solution schemes for the 2-D SWE have been presented in the literature and are claimed to be superior to the Leendertse scheme.

The University of Queensland developed an explicit 2-D hydrodynamic program (Apelt et al, 1974). This program is fairly limited and has been used only for research applications and was therefore given minor consideration. Also for reasons to be discussed later it was desirable to have an implicit or unconditionally stable 2-D scheme.

As the above 2-D hydrodynamic programs are not representative of the present level of world expertise, it was decided to investigate the possibility of developing new computer code. If it became apparent that this could not be achieved in the given timeframe, the TWODIM code would be used for developing the 2-D/1-D dynamic link. Consideration was also given to purchasing suitable software.

**1.4 **

**Study Objectives **

The main objective of the study was to develop a computer program capable of modelling long waves using 1-D and 2-D solution schemes dynamically linked together. As discussed above, it was also highly desirable to develop a new computer program for the 2-D component, noting that the development of such a program would be a major exercise in itself. Consideration was also to be given towards the development of a computer graphics system for 2-D models to enhance data entry and output.

After initial investigations and literature reviews the objectives were broadly defined as:

1a. Develop a new computer program based on one or more 2-D schemes described in the literature.

1b. In parallel to 1a above, develop a computer graphics system for 2-D models.

2. Research and develop an algorithm for dynamically linking the 1-D scheme of the ESTRY program with the 2-D scheme resulting from 1a above.

During the course of the project it became apparent that further research and development of improved methods for modelling wetting and drying of intertidal flats in a 2-D model, and for handling oblique water level boundaries on an orthogonal 2-D grid was advantageous. The latter was especially required for allowing maximum flexibility when locating the 2-D/1-D link interface.

**1.5 **

**Thesis Layout **

**Chapter 2 **

Chapter 2 contains a discussion of the preliminary investigations carried out so as to form a solid research basis and establish methodologies prior to software development. Literature reviews and general notions on the dynamic linking of 1-D and 2-D schemes are presented. Guidelines for selecting 1-D and 2-D schemes are set down.

Based on these guidelines, the suitability of the ESTRY program for the 1-D component is established. Several of the more prominent 2-D schemes presented in the literature are reviewed and assessed for the purpose of developing a new 2-D hydrodynamic program. One of these 2-D schemes is selected.

*INTRODUCTION*

**1-7**

**1-7**

**Chapter 3 **

The mathematical methods used in the new 2-D hydrodynamic program are presented in Chapter 3. The performance of the methods are discussed and illustrated through computer graphic output. Not presented in any detail is the considerable effort which went into data administration and the numerous features, such as error messages and flexibility in specifying boundary values, which make a program highly user-friendly and easy to apply in practice.

The mathematical solution, which is based on Stelling (1984), is an alternating direction implicit (ADI) finite difference method over an orthogonal grid. A brief description of the computation procedure and the treatment of model boundaries as proposed by Stelling is presented. The methodology used by the author for coding the scheme is detailed.

The more interesting results of the many test cases used to validate the 2-D scheme are illustrated and examined. The scheme's performance under different conditions is established.

To enhance the performance of the 2-D scheme in practical situations a new wetting and drying method is proposed. The results of test cases highlighting the performance of the method is presented.

**Chapter 4 **

The method proposed for linking dynamically the 1-D and 2-D schemes is described in Chapter 4. The logic behind selecting the types of open boundaries at the interface is discussed followed by the details of the transfer of data from one model to the other.

The results of two of a range of test cases, which were used to validate the method and assess its performance, are presented. These tests show a comparison of results for idealised situations which are modelled in both a 1-D and a 2-D/1-D form.

To allow maximum flexibility in locating the 2-D/1-D interface, investigations into calculating the flow across a line oblique to a 2-D orthogonal grid, and for improving the stability of water level boundary conditions along an oblique line were made. Methods to handle these situations are proposed and detailed. The results of a test case highlighting their performance are presented.

**2 **

**S**

**ELECTION OF **

**S**

**CHEMES**

**2.1 Summary **

A discussion on the preliminary investigations carried out to establish the basis for the research and development of the 2-D/1-D hydrodynamic program is presented.

Firstly some base considerations are addressed, namely, whether to use finite differences or finite elements, the definition of the Courant Number, and the merits of explicit and implicit schemes. Secondly, a literature review and some general aspects of dynamically linking 1-D and 2-D schemes are presented. Based on this, broad guidelines are established for the selection of the 1-D and 2-D SWE solution schemes.

As discussed earlier, the program ESTRY was preferred for the 1-D component primarily because of the availability of its computer code and its widespread application. The ESTRY program is briefly described and is shown to be suitable for implementation into the 2-D/1-D hydrodynamic program based on the guidelines.

The 2-D SWE scheme is addressed. A selection of the more suitable schemes are reviewed based on reports in the literature and private communication. The scheme described in Stelling (1984) was selected for the development of a new 2-D hydrodynamic modelling program on the grounds of a good level of documentation and high standard of performance.

**2.2 Base Considerations **

**2.2.1 Finite Differences versus Finite Elements **

Only finite difference methods (FDM) have been considered for the purposes of this project as opposed to other methods such as that of finite elements (FEM). The reason for this is that FDM have tended to dominate the literature in the free-surface hydrodynamic modelling field and are by far the most popular for use in practice. Also, the ESTRY solution scheme is a FDM, and a practical link to it would require using a FDM for the 2-D scheme.

The domination of the FDM originates from their simplicity making them the only real option when computers were far more limited. In more recent times the FEM has found greater application for large scale models, but is still cumbersome and more expensive for the finer detail models (Benque et al 1982b). For a program to be of wide application, the FDM is still the preferred option. This has become even more so with the use of curvilinear grids which provide the same benefit that the FEM offers, namely better geometrical representation of a model's boundaries.

**2.2.2 Courant Number/Computation Timestep **

The Courant Number, Cr, extends from the 'Courant-Friedrichs-Lewy (CFL) condition' (Abbott 1979). An implication of the CFL condition is that the stability of an explicit scheme is conditional on Cr being less than 1.0. Only approximate solutions will occur for Cr less than 1.0, while for Cr

*SELECTION OF SCHEMES*

**2-2**

**2-2**

greater than 1.0 the solution will bear little or no resemblance to the true solution, ie an unstable simulation. A true solution will occur if Cr is equal to 1.0.

Implicit schemes are generally not conditional on Cr being less than 1.0 and are referred to as unconditionally stable. The inaccuracy (phase error) of an implicit scheme, however, increases with increasing Cr. The degree of inaccuracy can be established for simple cases using phase portraits (Abbott 1979), but is difficult to ascertain in practice other than by comparing with simulations which used lower values of Cr.

In practice the Courant Number is used as a guide for calculating the computation timestep for a simulation. Assuming the mean water velocity to be small compared with the wave celerity, the formula for Cr is given by Equations 2.2.1 (1-D) and 2.2.2 (2-D). In the case of a 2-D square grid Equation 2.2.2 reduces to Equation 2.2.3.

*x*

*gH*

*t*

*=*

*Cr*

### ∆

### ∆

**1-D Scheme (2.2.26)**

*m*

* water,*

*of*

*depth*

*=*

*H*

*ms*

*gravity,*

*to*

*due*

*on*

*accelerati*

*=*

*g*

*m*

*element,*

*model*

*of*

*length*

*=*

*x*

* s*

*timestep,*

*=*

*t*

*where*

*2*

*1*

### -∆

### ∆

∆ ∆ ∆*y*

*1*

*+*

*x*

*1*

*gH*

*t*

*=*

*Cr*

*2*

*2*

*2*

*1*

**2-D Scheme (2.2.2)**

*x*

*2gH*

*t*

*=*

*Cr*

### ∆

### ∆

**2-D Square Grid (2.2.3)**

In reality the value of Cr varies over the computational domain, as either the element lengths and/or depths vary (the timestep usually being constant). Usually when a Courant Number is quoted it is the largest value over the model domain. This is referred to as the critical Courant Number as it is used to determine the most suitable computation timestep. For 2-D models of constant element size, the average depth is sometimes used.

All Courant Numbers quoted in this thesis are critical values for the models considered. For tidal simulations, the critical Courant Number was calculated on the basis of water depths at mid tide (at mean water level).

**2.2.3 Explicit versus Implicit Schemes **

Explicit schemes allow the calculation of results in one analytical step based entirely on parameter values from the previous time step. Implicit schemes use formulations incorporatin g two or more unknown time dependent parameter values. They therefore require the use of iterative procedures and matrix solution methods, and are more difficult to formulate.

Mathematical stability analyses generally show that explicit schemes are restricted to Courant Numbers less than unity which requires the use of small computation timesteps. Explicit schemes are, however, generally easier to formulate and use less computational effort per timestep than implicit schemes.

The advantage of using implicit schemes is that they are not restricted to Courant Numbers less than unity, and can therefore use larger timesteps than explicit schemes. Clearly there must be an upper limit to the timestep, the determination of which varies depending on the scheme and complexity of the flow. A too large a timestep is generally characterised by either instabilities and/or inaccuracies. For most implicit schemes in use today, a critical Courant Number between 2 and 15 is typically used.

The preference for an explicit or an implicit scheme is partly a personal issue dependent on the type of schemes the person is familiar with. Historically, implicit schemes were often preferred because of their capability to provide stability at large time steps, thereby reducing computational effort.

For 1-D schemes the issue of computation speed superiority of implicit schemes is less important with today's level of computing power except for extremely large models or simulations over a long period of time. Thus, with regard to the 1-D scheme, either an explicit or implicit method would suffice for this study.

For 2-D SWE schemes, explicit methods can be satisfactory for large grid sizes as the number of timesteps per tidal cycle is not prohibitive. An example given in Benque et al (1982b) for a model of the English Channel with a grid spacing of 10 km required 200 timesteps per tide. In comparison, a model with a grid spacing of 20 m would require in excess of 100,000 timesteps per tidal cycle which can be prohibitive depending on the computer hardware. As the 2-D scheme was to be as wide ranging as possible in its application, it was decided that an implicit 2-D scheme would be preferable to an explicit one.

**2.3 Linking 1-D and 2-D Schemes **

**2.3.1 General **

This section presents a general discussion on the dynamic linking of 1-D and 2-D schemes. Firstly a literature review is given followed by consideration of a number of key aspects as given below.

• methodology of 2-D/1-D dynamic link,

• timestep compatibility of 1-D and 2-D schemes,

• Courant number/stability criteria of 1-D and 2-D models,

• oblique orientation of 2-D/1-D interface to a 2-D orthogonal grid.

**2.3.2 Literature Review **

A review of Stelling (1981), Danish Hydraulic Institute (1983) and Crean et al (1988) with regard to the linking of 1-D and 2-D SWE schemes is given below.

*SELECTION OF SCHEMES*

**2-4**

**2-4**

**Stelling (1981) **

Stelling (1981) examines several methods for coupling or linking 1-D and 2-D schemes. The schemes used are both implicit with the 2-D scheme being based on that of Leendertse (1970).

**Figure 2.1 Spatial Grid of Coupled Model, (Figure 4.1, Stelling 1981)**

Three methods are proposed and are briefly described below. They are based on the prescription of internal boundary conditions at the interface of the two models in conjunction with the governing equations. The internal boundary conditions are based on a common water surface level and equivalence of flow in the direction perpendicular to the internal boundary of the 2-D model. The governing equations at the interface need only be expressed in a 1-D form in the direction of the flow. For the methods presented, Figure 2.1 illustrates the spatial grid of the coupled 1-D/2-D model arrangement. A virtual velocity point is created which is where the equivalence of flow condition is applied.

Method 1 Uses the virtual flow point and a one-sided difference form of the continuity equation. Method 2 As for Method 1 but the characteristic equations are used. In the case examined only

the outgoin g characteristic equation is used. The advantage of this method appears to be the capacity to overcome the one-sided differencing used in Method 1 by using either the out-going or in-going form of the equations as applicable.

Method 3 This method, which is different to the above two, uses an overlapping of the 1-D non-staggered grid with the 2-D non-staggered grid. The internal boundary conditions and forms of the continuity and momentum equations can then be applied at the overlap.

The accuracy of each method is assessed through simple model test cases. In each case spurious waves or numerical disturbances were generated as a wave passed through the internal boundaries of the two models. Method 2 was shown to produce by far the most accurate results. Stelling concludes that:

'A coupling between existing 1-D and 2-D models can be established indeed. Coupling schemes based upon characteristics are likely to give the most accurate results.'

**Danish Hydraulic Institute (1983) **

A coupling of the Danish Hydraulic Institute S11 (1-D scheme) and S21 (2-D scheme) computer programs in an application to the Gambia estuary and river is given in Danish Hydraulic Institute (1983). No detail or reference is provided other than that the 'coupling is led through a special component that is comprised of a set of parallel one-dimensional channels with constraints imposed upon lateral exchanges between them'.

**Crean et al (1988) **

Crean et al (1988) describes the development of a combined 1-D and 2-D model codenamed GF2 of the Strait of Georgia, Juan de Fuca Strait, and adjoining passages and inlets. The 2-D scheme is solved explicitly over a square mesh. Two 1-D schemes are used, the first being explicit and an adaptation of the 2-D scheme, the second being implicit. The 1-D explicit scheme does not include the advective accelerations.

At each junction the width of the channel in the 1-D model is assumed to be the same as the 2-D square mesh size. To obtain the correct cross-sectional area the depth of the 1-D channel is adjusted. The linking of the 2-D and explicit 1-D schemes was carried out as follows. When solving the 1-D continuity equation for the water level adjacent to the 2-D model, the adjoining 2-D grid element is treated as an extension to the 1-D model. This allows the flow into/out of the 1-D model to be defined. For calculating the boundary water level of the 2-D model a Lagrangian interpolation of the adjacent water levels in both the 1-D and 2-D models was used. This appears to be necessary as the spatial location of the 1-D and 2-D water level boundary values are not coincident.

For linking the 2-D and implicit 1-D schemes a similar approach was adopted, but the need for the Lagrangian interpolation is removed by defining the length of the first element of the 1-D model as equal to the mesh size of the 2-D model. This allows the 2-D continuity equation to be used at the boundary of the 2-D model by adopting the first element of the 1-D model to be an extension of the 2-D model in a similar manner to that used for the 1-D model as described above.

**2.3.3 Comments on the Literature **

The three sources listed above all present or illustrate one or more methods for linking 1-D and 2-D schemes. Comments on these methods and their suitability are presented below:

(i). All of the methods appear to be limited in that only one 2-D grid element can be connected to a 1-D model as opposed to any number of 2-D elements. This problem is overcome to some extent in the Danish Hydraulic Institute example where the 1-D model is branched into a number of parallel channels with each of these connecting to a single 2-D grid element. Such a procedure is, however, somewhat cumbersome.

*SELECTION OF SCHEMES*

**2-6**

**2-6**

literature and is therefore uncertain. Some concern must be expressed over the occurrence of the numerical disturbances and instabilities mentioned.

(iii). The methods described in Crean et al (1988) appear limited because of the restrictions on the linkage of 1-D and 2-D schemes.

(iv). The use of two different solution schemes for the 1-D and 2-D models is plausible.

**2.3.4 Methodology of 2-D/1-D Dynamic Link **

There are essentially two types of methodologies that can be used for linking 1-D and 2-D schemes. They are:

1. The 1-D and 2-D schemes are solved independently, with each scheme "feeding" the other boundary values after each computation step. Such an arrangement allows the 1-D and 2-D schemes to be autonomous. However, both schemes must have "non-reflective" characteristics at an open boundary so the conveyance of mass from one model to another is accurately represented.

2. The 1-D and 2-D schemes use a similar solution technique so that both the 1-D and 2-D equations are solved simultaneously. In this situation, the 1-D sections would essentially be an extension of the 2-D grid.

Most of the methods described in Section 2.3.2 are of the first type. Variations on this methodology are used however, such as overlapping models so they have common computation points or the use of virtual points outside the model domains.

The first type was selected in this project in preference to the second as it allowed greater freedom when selecting the 1-D and 2-D schemes and has potentially more flexibility and a wider range of applicability. Limitations may be "reflective" boundaries at the 2-D/1-D interface and harsher stability constraints.

**2.3.5 Computation Timestep **

To avoid extrapolation of boundary values at the 2-D/1-D interface, it is necessary for the computation timestep to be the same for both schemes. Extrapolation of boundary values would require an iterative procedure to ensure convergence occurred. This is unlikely to provide any benefits in computation efficiency.

A common computation timestep for both the 1-D and 2-D schemes implies that one of the schemes would be running inefficiently as it would be using a smaller timestep than that which would be used to run the scheme on its own.

**2.3.6 Courant Number Constraints **

By adopting a common computation timestep, the critical Courant Number of each of the 1-D and 2-D models would differ greatly, with the value for a 2-2-D model being typically greater than that of a 1-D model because of the finer or smaller spatial discretisation in the 2-1-D model. Adopting this as being the norm, the issue of whether the solution schemes should be explicit or implicit is discussed

below. For the purposes of the discussion, an explicit scheme implies the critical Courant Number is less than unity, while for an implicit scheme a value no greater than fifteen is implied.

If both the 1-D and D schemes are implicit the computation timestep would be controlled by the 2-D model. This necessitates that the simulation of the 1-2-D model would be computationally inefficient because of the comparatively short timestep. However, this would generally be of minor consequence as the majority of computational effort would be for the 2-D model. With both schemes being implicit this arrangement offers the maximum flexibility and is preferred for this reason. The same logic applies if both schemes are explicit. However, an explicit 2-D scheme can be limited in its practical application because of the large computation requirements when modelling with small grid lengths (Section 2.2.3).

An implicit 2-D scheme linked to an explicit 1-D scheme is also practicable. Where the timestep is controlled by the 2-D solution, this is the optimum combination, as an explicit 1-D scheme is more computationally efficient per timestep than an implicit one. This is, however, of only minor benefit unless the computational effort for the 1-D model is significant in comparison to that for the 2-D model. Where the timestep is controlled by the explicit 1-D scheme, however, this arrangement would be computationally inefficient.

An explicit 2-D scheme / implicit 1-D scheme combination would be limiting except in the rare situation where the Courant Number for the 2-D scheme is much lower than that of the 1-D scheme. The above implies that the order of preference for the schemes would be:

1. Implicit 2-D / Implicit 1-D 2. Implicit 2-D / Explicit 1-D 3. Explicit 2-D / Explicit 1-D 4. Explicit 2-D / Implicit 1-D

The first two arrangements are much preferred over the last two as it was desirable not to have an explicit 2-D scheme as discussed previously.

The arrangement of schemes finally adopted was the implicit 2-D / explicit 1-D combination.

**2.3.7 Length and Orientation of 2-D/1-D Interface **

The methods presented in the literature for linking 1-D and 2-D schemes (Section 2.3.2) appear to allow only the connection of one 2-D grid element to one 1-D element for each link. This is restrictive as it would clearly be desirable for the 2-D/1-D interface to extend over more than one 2-D element as shown in Figure 2.2.

There is also the problem that for an orthogonal 2-D grid difficulty can occur in representing accurately an open boundary along a line which is oblique to the X and Y axes of the grid. This is due to the irregularity of the boundary line as boundary values can only be specified at fixed grid points. It would therefore be necessary to test the accuracy and performance of open boundaries

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would be significant practical benefits if a 2-D/1-D interface can be located at an oblique boundary as is illustrated in Figure 2.3.

**Figure 2.2 A 1-D Element Linked to **
**Multiple 2-D Grid Elements **

**Figure 2.3 2-D/1-D Link at an Oblique **
**Boundary **

**2.4 1-D Scheme of ESTRY **

**2.4.1 General **

ESTRY, which has been developed by WBM Pty Ltd since 1974, is a proven and established program for modelling floods and tides using 1-D network schematisations. During ESTRY's development, it has been subject to validation testing (Collins 1988) and scrutinised by WBM employee's and others. It has been shown to be reliable and accurate for a wide range of tidal and flood applications, as demonstrated by calibration and verification to recorded level and flow data, and by rigorous assessment in various "test case" situations.

Because of ESTRY's recognition, capabilities and availability, it was decided to incorporate its code into the 2-D/1-D program.

**2.4.2 Description **

ESTRY, by default, uses an explicit finite difference, second-order, Runge-Kutta solution technique (Morrison and Smith, 1978) for the 1-D SWE of continuity and momentum as given by Equations 2.4.1 and 2.4.2. An implic it scheme was also developed, however, testing and experience has shown the explicit scheme to be preferred for production simulations. The equations contain the essential terms for modelling periodic long waves in estuaries and rivers, that is: wave propagation; advection of momentum (inertia terms) and bed friction (Manning's equation).

*0*

*=*

*t*

*B*

*+*

*x*

*(uA)*

### ∂

### ∂

### ∂

### ∂

*ζ*

**(2.4.1)**

*0*

*=*

*u *

* |*

*u*

*| *

*k*

*+*

*x*

*g*

*+*

*x*

*u*

*u*

*+*

*t*

*u*

### ∂

### ∂

### ∂

### ∂

### ∂

### ∂

*ζ*

**(2.4.2)**

*gravity*

*to*

*due*

*on*

*accelerati*

*=*

*g*

*Radius*

*Hydraulic*

*=*

*R*

*n*

*annings*

*M*

*=*

*n*

*R*

*gn*

*=*

*t*

*coefficien*

*loss*

*energy*

*=*

*k*

*flow*

*of*

*width*

*=*

*B*

*area*

*sectional*

*cross*

*=*

*A*

*distance*

*=*

*x*

*time*

*=*

*t*

*level*

*water*

*=*

*velocity*

*averaged*

*width*

*and*

*depth*

*=*

*u*

*where*

*4/3*

*2*′

*ζ*

The spatial discretisation of an area of interest is carried out as a network of interconnected nodes and channels. The nodes represent the storage or pondage characteristics of a river, floodplain or estuary, while the channels model the hydraulic conveyance characteristics. An example of an ESTRY network was shown in Figure 1.1 of Chapter 1.

Nodes are relatively simple to construct. Each node is defined as a table of "still water" surface areas at selected elevations. From this, the volume or storage capacity of the river or estuary is defined. Intermediate values are interpolated linearly.

The channels represent the flow conveyance paths between the nodes. They are defined, in their simplest form, as a table of flow width versus elevation. In more complex forms, the cross-sectional area, wetted perimeter and/or Manning's n are specified as a function of the elevation to allow more appropriate calculation of the channel conveyance.

Channels also require details such as length, Manning's n value, and form loss coefficient (a dynamic head loss factor). A channel maybe specified as either a gradient channel, weir, bridge, culvert or the default which is a conventional channel. Gradient channels are designed for overbank flowpaths or where steep water gradients occur. They require an upstream and downstream channel bed level and include an algorithm for free-overfall such as which occurs when flood waters return to a river across a levee as the river waters recede. Weir and bridge channels have essentially zero length. Algorithms for these channels are based on standard engineering design publications and practices. Allowances

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are made for submergence of bridge decks and downstream control for flow over a weir. The culvert algorithms, which represent all flow regimes, are based on standard formulae and have been tested successfully against manufacturers standards.

The continuity equation is solved at the nodes, while the momentum equation is solved for conventional and gradient channels. The output consists of water levels at the nodes, and flows, velocities and integral flows (flow integrated over time) at the channels.

All boundary conditions for an ESTRY model are applied at the nodes. These may be a constant flow, flow versus time, flow versus water level, harmonic water level, water level versus time or water level versus flow (a stage-discharge curve). Flow boundary values are applied at nodes when the continuity equation is solved, while water level boundary values are used during the solution of the momentum equation.

**2.4.3 Suitability **

The reasons for and against using ESTRY for the 1-D scheme are discussed below.

4 The main possible drawback of using ESTRY was that its explicit scheme requires the Courant Number to be less than unity. As discussed in Section 2.3.6 an explicit 1-D scheme is

acceptable, and even preferred where the timestep is controlled by the 2-D scheme but is limiting where it is controlled by the 1-D scheme. Careful consideration was given to this, upon which it was concluded that in the majority of cases:

(a) The timestep would be controlled by the 2-D scheme because of its finer spatial resolution. (b) The area of interest would lie in the 2-D model, so that if the ESTRY explicit scheme was

imposing a limitation on the timestep, the spatial resolution in the constraining areas of the 1-D model could be made coarser, thus allowing a longer timestep without any likely loss of accuracy in the 2-D model.

An explicit 1-D scheme was therefore not considered to be a limitation in practice.

5 ESTRY is well suited in terms of its application of boundary conditions as both water level and flow boundaries are located at nodes. This simplifies to some extent any problems associated with the spatial location of water level and flow points at the 2-D/1-D interface.

6 There are of course the benefits already discussed of using a scheme which is widely accepted and proven in practice.

On the basis of the above it was decided to adopt the 1-D scheme of ESTRY.

**2.5 Two-Dimensional Scheme Selection **

**2.5.1 Summary **

The majority of 2-D schemes which have found widespread practical application use the alternating direction implicit (ADI) FDM. Other schemes have utilised finite elements FEM, the fractional step approach, explicit FDM and fully implicit FDM.